Average Error: 34.4 → 10.4
Time: 4.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r81182 = b_2;
        double r81183 = -r81182;
        double r81184 = r81182 * r81182;
        double r81185 = a;
        double r81186 = c;
        double r81187 = r81185 * r81186;
        double r81188 = r81184 - r81187;
        double r81189 = sqrt(r81188);
        double r81190 = r81183 - r81189;
        double r81191 = r81190 / r81185;
        return r81191;
}
↓
double f(double a, double b_2, double c) {
        double r81192 = b_2;
        double r81193 = -1.0674124610604968e-82;
        bool r81194 = r81192 <= r81193;
        double r81195 = -0.5;
        double r81196 = c;
        double r81197 = r81196 / r81192;
        double r81198 = r81195 * r81197;
        double r81199 = 5.968766258400916e+107;
        bool r81200 = r81192 <= r81199;
        double r81201 = -r81192;
        double r81202 = cbrt(r81201);
        double r81203 = r81202 * r81202;
        double r81204 = r81192 * r81192;
        double r81205 = a;
        double r81206 = r81205 * r81196;
        double r81207 = r81204 - r81206;
        double r81208 = sqrt(r81207);
        double r81209 = -r81208;
        double r81210 = fma(r81203, r81202, r81209);
        double r81211 = r81210 / r81205;
        double r81212 = 0.5;
        double r81213 = r81212 * r81197;
        double r81214 = 2.0;
        double r81215 = r81192 / r81205;
        double r81216 = r81214 * r81215;
        double r81217 = r81213 - r81216;
        double r81218 = r81200 ? r81211 : r81217;
        double r81219 = r81194 ? r81198 : r81218;
        return r81219;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -1.0674124610604968e-82

    1. Initial program 52.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 8.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -1.0674124610604968e-82 < b_2 < 5.968766258400916e+107

    1. Initial program 13.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt13.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}\right) \cdot \sqrt[3]{-b_2}} - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    4. Applied fma-neg13.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]

    if 5.968766258400916e+107 < b_2

    1. Initial program 50.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 3.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))